H-Termination proof of /home/matraf/haskell/eval_FullyBlown

YES 0.721 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/empty.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ LR

mainModule Main

((sequence_ :: Monad b => [b a] -> b ()) :: Monad b => [b a] -> b ())

module Main where

import qualified Prelude

Lambda Reductions:
The following Lambda expression

\_→q

is transformed to

gtGt0 q _ = q

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule Main

((sequence_ :: Monad a => [a b] -> a ()) :: Monad a => [a b] -> a ())

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Main

((sequence_ :: Monad b => [b a] -> b ()) :: Monad b => [b a] -> b ())

module Main where

import qualified Prelude

Cond Reductions:
The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

mainModule Main

(sequence_ :: Monad a => [a b] -> a ())

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="sequence_\n  Monad a",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="sequence_ vy3\n  Monad a",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldr (>>) (return ()) vy3\n  Monad a",fontsize=16,color="burlywood",shape="triangle"];62[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 5[label="",style="solid", color="burlywood", weight=3];
63[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldr (>>) (return ()) (vy30 : vy31)\n  Monad a",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldr (>>) (return ()) []\n  Monad a",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="(>>) vy30 foldr (>>) (return ()) vy31\n  Monad a",fontsize=16,color="blue",shape="box"];64[label=">> :: (IO a) -> (IO ()) -> IO ()",fontsize=10,color="white",style="solid",shape="box"];7 -> 64[label="",style="solid", color="blue", weight=9];
64 -> 21[label="",style="solid", color="blue", weight=3];
65[label=">> :: ([] a) -> ([] ()) -> [] ()",fontsize=10,color="white",style="solid",shape="box"];7 -> 65[label="",style="solid", color="blue", weight=9];
65 -> 22[label="",style="solid", color="blue", weight=3];
66[label=">> :: (Maybe a) -> (Maybe ()) -> Maybe ()",fontsize=10,color="white",style="solid",shape="box"];7 -> 66[label="",style="solid", color="blue", weight=9];
66 -> 23[label="",style="solid", color="blue", weight=3];
8[label="return ()\n  Monad a",fontsize=16,color="blue",shape="box"];67[label="return :: () -> IO ()",fontsize=10,color="white",style="solid",shape="box"];8 -> 67[label="",style="solid", color="blue", weight=9];
67 -> 11[label="",style="solid", color="blue", weight=3];
68[label="return :: () -> [] ()",fontsize=10,color="white",style="solid",shape="box"];8 -> 68[label="",style="solid", color="blue", weight=9];
68 -> 12[label="",style="solid", color="blue", weight=3];
69[label="return :: () -> Maybe ()",fontsize=10,color="white",style="solid",shape="box"];8 -> 69[label="",style="solid", color="blue", weight=9];
69 -> 13[label="",style="solid", color="blue", weight=3];
21 -> 15[label="",style="dashed", color="red", weight=0];
21[label="(>>) vy30 foldr (>>) (return ()) vy31",fontsize=16,color="magenta"];21 -> 28[label="",style="dashed", color="magenta", weight=3];
22 -> 16[label="",style="dashed", color="red", weight=0];
22[label="(>>) vy30 foldr (>>) (return ()) vy31",fontsize=16,color="magenta"];22 -> 29[label="",style="dashed", color="magenta", weight=3];
23 -> 17[label="",style="dashed", color="red", weight=0];
23[label="(>>) vy30 foldr (>>) (return ()) vy31",fontsize=16,color="magenta"];23 -> 30[label="",style="dashed", color="magenta", weight=3];
11[label="return ()",fontsize=16,color="black",shape="box"];11 -> 18[label="",style="solid", color="black", weight=3];
12[label="return ()",fontsize=16,color="black",shape="box"];12 -> 19[label="",style="solid", color="black", weight=3];
13[label="return ()",fontsize=16,color="black",shape="box"];13 -> 20[label="",style="solid", color="black", weight=3];
28 -> 4[label="",style="dashed", color="red", weight=0];
28[label="foldr (>>) (return ()) vy31",fontsize=16,color="magenta"];28 -> 37[label="",style="dashed", color="magenta", weight=3];
15[label="(>>) vy30 vy4",fontsize=16,color="black",shape="triangle"];15 -> 24[label="",style="solid", color="black", weight=3];
29 -> 4[label="",style="dashed", color="red", weight=0];
29[label="foldr (>>) (return ()) vy31",fontsize=16,color="magenta"];29 -> 38[label="",style="dashed", color="magenta", weight=3];
16[label="(>>) vy30 vy4",fontsize=16,color="black",shape="triangle"];16 -> 25[label="",style="solid", color="black", weight=3];
30 -> 4[label="",style="dashed", color="red", weight=0];
30[label="foldr (>>) (return ()) vy31",fontsize=16,color="magenta"];30 -> 39[label="",style="dashed", color="magenta", weight=3];
17[label="(>>) vy30 vy4",fontsize=16,color="black",shape="triangle"];17 -> 26[label="",style="solid", color="black", weight=3];
18[label="primretIO ()",fontsize=16,color="black",shape="box"];18 -> 27[label="",style="solid", color="black", weight=3];
19[label="() : []",fontsize=16,color="green",shape="box"];20[label="Just ()",fontsize=16,color="green",shape="box"];37[label="vy31",fontsize=16,color="green",shape="box"];24[label="vy30 >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];24 -> 31[label="",style="solid", color="black", weight=3];
38[label="vy31",fontsize=16,color="green",shape="box"];25[label="vy30 >>= gtGt0 vy4",fontsize=16,color="burlywood",shape="triangle"];76[label="vy30/vy300 : vy301",fontsize=10,color="white",style="solid",shape="box"];25 -> 76[label="",style="solid", color="burlywood", weight=9];
76 -> 32[label="",style="solid", color="burlywood", weight=3];
77[label="vy30/[]",fontsize=10,color="white",style="solid",shape="box"];25 -> 77[label="",style="solid", color="burlywood", weight=9];
77 -> 33[label="",style="solid", color="burlywood", weight=3];
39[label="vy31",fontsize=16,color="green",shape="box"];26[label="vy30 >>= gtGt0 vy4",fontsize=16,color="burlywood",shape="box"];78[label="vy30/Nothing",fontsize=10,color="white",style="solid",shape="box"];26 -> 78[label="",style="solid", color="burlywood", weight=9];
78 -> 34[label="",style="solid", color="burlywood", weight=3];
79[label="vy30/Just vy300",fontsize=10,color="white",style="solid",shape="box"];26 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 35[label="",style="solid", color="burlywood", weight=3];
27[label="terminator ()",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
31[label="primbindIO vy30 (gtGt0 vy4)",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3];
32[label="vy300 : vy301 >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];32 -> 41[label="",style="solid", color="black", weight=3];
33[label="[] >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];33 -> 42[label="",style="solid", color="black", weight=3];
34[label="Nothing >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];34 -> 43[label="",style="solid", color="black", weight=3];
35[label="Just vy300 >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];35 -> 44[label="",style="solid", color="black", weight=3];
36[label="ter0m ()",fontsize=16,color="green",shape="box"];36 -> 45[label="",style="dashed", color="green", weight=3];
40[label="terminator vy30 (gtGt0 vy4)",fontsize=16,color="black",shape="box"];40 -> 46[label="",style="solid", color="black", weight=3];
41 -> 47[label="",style="dashed", color="red", weight=0];
41[label="gtGt0 vy4 vy300 ++ (vy301 >>= gtGt0 vy4)",fontsize=16,color="magenta"];41 -> 48[label="",style="dashed", color="magenta", weight=3];
42[label="[]",fontsize=16,color="green",shape="box"];43[label="Nothing",fontsize=16,color="green",shape="box"];44[label="gtGt0 vy4 vy300",fontsize=16,color="black",shape="box"];44 -> 49[label="",style="solid", color="black", weight=3];
45[label="()",fontsize=16,color="green",shape="box"];46[label="ter1m vy30 (gtGt0 vy4)",fontsize=16,color="green",shape="box"];46 -> 50[label="",style="dashed", color="green", weight=3];
46 -> 51[label="",style="dashed", color="green", weight=3];
48 -> 25[label="",style="dashed", color="red", weight=0];
48[label="vy301 >>= gtGt0 vy4",fontsize=16,color="magenta"];48 -> 52[label="",style="dashed", color="magenta", weight=3];
47[label="gtGt0 vy4 vy300 ++ vy5",fontsize=16,color="black",shape="triangle"];47 -> 53[label="",style="solid", color="black", weight=3];
49[label="vy4",fontsize=16,color="green",shape="box"];50[label="vy30",fontsize=16,color="green",shape="box"];51[label="gtGt0 vy4",fontsize=16,color="grey",shape="box"];51 -> 54[label="",style="dashed", color="grey", weight=3];
52[label="vy301",fontsize=16,color="green",shape="box"];53[label="vy4 ++ vy5",fontsize=16,color="burlywood",shape="triangle"];82[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];53 -> 82[label="",style="solid", color="burlywood", weight=9];
82 -> 55[label="",style="solid", color="burlywood", weight=3];
83[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];53 -> 83[label="",style="solid", color="burlywood", weight=9];
83 -> 56[label="",style="solid", color="burlywood", weight=3];
54[label="gtGt0 vy4 vy6",fontsize=16,color="black",shape="triangle"];54 -> 57[label="",style="solid", color="black", weight=3];
55[label="(vy40 : vy41) ++ vy5",fontsize=16,color="black",shape="box"];55 -> 58[label="",style="solid", color="black", weight=3];
56[label="[] ++ vy5",fontsize=16,color="black",shape="box"];56 -> 59[label="",style="solid", color="black", weight=3];
57[label="vy4",fontsize=16,color="green",shape="box"];58[label="vy40 : vy41 ++ vy5",fontsize=16,color="green",shape="box"];58 -> 60[label="",style="dashed", color="green", weight=3];
59[label="vy5",fontsize=16,color="green",shape="box"];60 -> 53[label="",style="dashed", color="red", weight=0];
60[label="vy41 ++ vy5",fontsize=16,color="magenta"];60 -> 61[label="",style="dashed", color="magenta", weight=3];
61[label="vy41",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_psPs(:(vy40, vy41), vy5) → new_psPs(vy41, vy5)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

new_psPs(:(vy40, vy41), vy5) → new_psPs(vy41, vy5)
The graph contains the following edges 1 > 1, 2 >= 2

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPSizeChangeProof

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_gtGtEs(:(vy300, vy301), vy4, ba) → new_gtGtEs(vy301, vy4, ba)

From the DPs we obtained the following set of size-change graphs:

new_gtGtEs(:(vy300, vy301), vy4, ba) → new_gtGtEs(vy301, vy4, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

new_foldr(:(vy30, vy31), ty_[], ba) → new_foldr(vy31, ty_[], ba)
new_foldr(:(vy30, vy31), ty_IO, ba) → new_foldr(vy31, ty_IO, ba)
new_foldr(:(vy30, vy31), ty_Maybe, ba) → new_foldr(vy31, ty_Maybe, ba)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                          ↳ QDPSizeChangeProof

                        ↳ QDP

                        ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_foldr(:(vy30, vy31), ty_Maybe, ba) → new_foldr(vy31, ty_Maybe, ba)

From the DPs we obtained the following set of size-change graphs:

new_foldr(:(vy30, vy31), ty_Maybe, ba) → new_foldr(vy31, ty_Maybe, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ QDPSizeChangeProof

                        ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

new_foldr(:(vy30, vy31), ty_IO, ba) → new_foldr(vy31, ty_IO, ba)

From the DPs we obtained the following set of size-change graphs:

new_foldr(:(vy30, vy31), ty_IO, ba) → new_foldr(vy31, ty_IO, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                        ↳ QDP

                          ↳ QDPSizeChangeProof

Q DP problem:
The TRS P consists of the following rules:

new_foldr(:(vy30, vy31), ty_[], ba) → new_foldr(vy31, ty_[], ba)

From the DPs we obtained the following set of size-change graphs:

new_foldr(:(vy30, vy31), ty_[], ba) → new_foldr(vy31, ty_[], ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3